Statistics in Retail Finance. Chapter 7: Profit estimation

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1 Statistics in Retail Finance 1

2 Overview > In this chapter we cover various methods to estimate profits at both the account and aggregate level based on the dynamic behavioural models introduced in the previous chapter:- Dynamic profit estimates using Markov transition models. Lifetime profit estimation using survival models. 2

3 Introduction > So far our interest has been in models of default. But why model default, when ultimately our interest in in profit? There is now great interest in moving to profit modelling in retail banks. A default model is a risk model. A profit model is a model of risk and reward. When producing profit models, we can give account-level and/or aggregate estimates (ie across a portfolio of loans). 3

4 Profit calculation using Markov transition model > In this section, we consider using underlying Markov transition models as the basis of profit calculations to rectify this problem. This approach is particularly suited to revolving credit where there is no fixed term of loan repayments. We need to define some terms: Suppose we model states for loans. is the probability of moving from state to state in time period, as estimated by a first-order Markov transition model. is the state at period : assume known. is profit associated with being in state. 4

5 In this setting, we define profit recursively: is the expected value of profit at period after the last periods, given an initial state of at period. Therefore { Notice that this calculation requires summing recursively over the probability transitions from period to. Expected profit across the whole period to is given by. It is computed as a recursive function. 5

6 Example 7.1 Suppose,,, and a stationary transition matrix. Compute expected profit up to and including time period. Solution [ ] [ ] [ ] [ ] [ ] 6

7 Introducing the default event > Define an indicator of default and a default cost: { is default cost. Then extend the expected profit calculation to { [ ] The principle here is that default is a one-off cost. It is only included in the model if the previous state was not a default. This is what the indicator does. 7

8 Example 7.2 Suppose we have two risk grades (1,2) and a default state (3). The transition matrix between states is represented (for all time periods) by The profits associated with risk grades 1 and 2 are 10 and 8 respectively. The cost of default is 200. Time periods are in months. Calculations of expected profit for different values are given in this table:- Therefore, if we suppose an initial state 2, then expected profits after 24 months are

9 Aggregate estimates > Aggregate expected profit can be computed by a weighted sum over expected profits computed for each state. So if the number of accounts with initial state is given by expected value of aggregate profit at period is given by then the 9

10 Exercise 7.1 Consider a two state stationary Markov transition model with transition matrix as a model of movers, accounts that are less likely to stay on one state than to move into another: more precisely, and. Let and where ; ie state 2 incurs a loss. 1. Show that. 2. If, show that expected profit is always positive: for all and states and. Do not consider default in the profit calculations. 10

11 Lifetime profit calculation > Profit can be calculated more systematically through the lifetime of a loan. Survival models can then be used to estimate risks within the context of lifetime profit calculations. We begin by considering profit calculations without default. Then, later default events are introduced and expected profit calculations are made. 11

12 Some accounting terms > We introduce some new definitions for fixed term loans. is number of periods of the loan (typically in months). is interest rate per period. o Note o If is annual interest rate and period is monthly then. is the risk-free rate per period. o This is the return that the lender could have got with no risk to capital, on the same loan amount (eg US government bonds), so raw profits should be adjusted by this rate. is loan value at period for { }. o Since the loan is fully paid back by period,. is repayment amount at time. 12

13 Calculating profit without default > Profits are calculated either as raw monetary profit,, or as net present value (NPV),, when profit is adjusted by the risk-free rate: and The development of the loan in each period is expressed as Therefore. 13

14 Suppose we choose a fixed repayment amount per term, so for all { }. Then And, in particular, for, Then and [ [ ] ] 14

15 Introducing the default event > Let be the probability that payments have been made up to period t. Then profitability calculation can be expanded to include this probability to get an expected value of profit: Additionally, the possible amount recovered needs to be factored in. Expected recovery = PD (1-LGD) EAD. Then probability of default (PD) specifically in period is the probability that payments have been made up to that period times hazard of default at that period:. is loss given default (LGD). Exposure at default (EAD) is given by. 15

16 This gives the expected value of profit [ ] Take a fixed repayment amount and substitute the formula for : [ ( )] [ ( ( ) )] [ [ ( )] ] 16

17 The expected value of NPV can also be derived in a similar way: [ [ ( ) ] ] 17

18 Profit calculation using the survival model > Including survival probability (for default) in lifetime profit model. Use lifetime profit calculation with default. is time within which to measure defaults over missed payments. o Eg if default is defined as three months consecutive missed payments, then. Then, given a default survival model. The hazard function can be the usual instantaneous hazard or, since the period is not an instant, use the hazard probability given by. 18

19 Example 7.3 A Cox PH model is run on a training data set of personal loans with predictor variables: age, employment status, tenure and months at address. Profits over a 12 month term can be calculated for any individual based on the model s survival probability. One individual is a 42 years old, employed home owner with 3 months in current address. 19

20 Survival probabilities over are shown in the following graph for this individual: Based on this survival function, an expected profit rate can be computed, given an interest rate and LGD

21 Extending the profit formulae > The profit formulae expressed here are given specifically for fixed term loans and where repayment amounts are fixed. However, it is feasible to extend them to unfixed terms, revolving credit or other repayment regimes (eg exponentially increasing amounts). In this model we are assuming LGD is fixed, but this may not be true and we may have a model for LGD too. In particular, LGD may well be correlated with PD. However, an LGD model could easily be incorporated in this specification. We have assumed fixed account details. However, these could easily be treated as time-varying by indexing by time period (ie ). This would fit in neatly with the use of time-varying covariates in survival modelling. 21

22 o In particular, this would enable the use of macroeconomic variables which would allow adjustment of profit forecasts based on forecasts of changes in the economy. Survival models of attrition and prepayment can also be incorporated in a similar way to the default model. 22

23 Exercise 7.2 Specify the profit formula based on using two survival functions: one which models default and another which models early account closure. Note that early account closure implies full prepayment of the outstanding loan. 23

24 Review of Chapter 7 > We covered profit estimation based on the following methods:- Dynamic profit estimates using Markov transition models. Lifetime profit estimation using survival models. 24